Descartes' Circle Formula

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(based on wording of ARML 2010 Power)

Descartes' Circle Formula is a relation held between four mutually tangent circles.

Some notation: when discussing mutually tangent circles (or arcs), it is convenient to refer to the curvature of a circle rather than its radius. We define curvature as follows. Suppose that circle A of radius $r_a$ is externally tangent to circle B of radius $r_b$. Then the curvatures of the circles are simply the reciprocals of their radii, $\frac{1}{r_a}$ and $\frac{1}{r_b}$.

If circle $A$ is internally tangent to circle $B$, however, a the curvature of circle $A$ is still $\frac{1}{r_a}$, while the curvature of circle B is $-\frac{1}{r_b}$, the opposite of the reciprocal of its radius.

$[asy] size(200); defaultpen(linewidth(0.7)); draw(Circle(origin,0.5)); draw(Circle((1.5,0),1)); dot(origin^^(1.5,0)^^(0.5,0)); draw(origin--(1.5,0)); label("1/2", (0.25,0), N); label("1", (1,0), N); label("A", origin, SW); label("B", (1.5,0), SE); [/asy]$

In the above diagram, the curvature of circle $A$ is $2$ while the curvature of circle $B$ is $1$.

$[asy] size(150); defaultpen(linewidth(0.7)); draw(Circle((1.25,0),0.25)); draw(Circle((1.5,0),0.5)); dot((1,0)^^(1.5,0)^^(1.25,0)^^(2,0)); draw((1,0)--(2,0)); label("1/2", (1.125,0), N); label("1", (1.75,0), N); label("A", (1.25,0), SW); label("B", (1.5,0), SE); [/asy]$

In the above diagram, the curvature of circle $A$ is still $2$ while the curvature of circle $B$ is $-1$.

When four circles $A, B, C,$ and $D$ are pairwise tangent, with respective curvatures $a, b, c,$ and $d$, then the following equation holds:

$(a+b+c+d)^2 = 2(a^2+b^2+c^2+d^2)$.